Theorem reu5 2825

Description: Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
reu5	|- (E!x e. A ph <-> (E.x e. A ph /\ E*x e. A ph))

Proof of Theorem reu5

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
2		df-reu 2622	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
3		df-rex 2621	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-rmo 2623	. . 3 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
5	3, 4	anbi12i 678	. 2 |- ((E.x e. A ph /\ E*x e. A ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
6	1, 2, 5	3bitr4i 268	1 |- (E!x e. A ph <-> (E.x e. A ph /\ E*x e. A ph))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   e. wcel 1710  E!weu 2204  E*wmo 2205  E.wrex 2616  E!wreu 2617  E*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  reurex  2826  reurmo  2827  reu4  3031  reueq  3034  fncnv  5159